Related papers: Quantum Approximation I. Embeddings of Finite Dime…
A basic problem of approximation theory, the approximation of functions from the Sobolev space W_p^r([0,1]^d) in the norm of L_q([0,1]^d), is considered from the point of view of quantum computation. We determine the quantum query…
We use entropy numbers in combination with the polynomial method to derive a new general lower bound for the n-th minimal error in the quantum setting of information-based complexity. As an application, we improve some lower bounds on…
Large-scale eigenvalue problems arise in various fields of science and engineering and demand computationally efficient solutions. In this study, we investigate the subspace approximation for parametric linear eigenvalue problems, aiming to…
Quantum counting is the task of determining the dimension of the subspace of states that are accepted by a quantum verifier circuit. It is the quantum analog of counting the number of valid solutions to NP problems -- a problem well-studied…
Quantum computers hold promise to enable efficient simulations of the properties of molecules and materials; however, at present they only permit ab initio calculations of a few atoms, due to a limited number of qubits. In order to harness…
This paper studies the probabilistic function approximation problem over reproducing kernel Hilbert spaces. We show the existence and uniqueness of the optimizer under mild assumptions. Furthermore, we generalize the celebrated representer…
Quantum statistical models (i.e., families of normalized density matrices) and quantum measurements (i.e., positive operator-valued measures) can be regarded as linear maps: the former, mapping the space of effects to the space of…
We study integration and $L_2$-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh,…
Motivated by recent progress in quantum technologies and in particular quantum software, research and industrial communities have been trying to discover new applications of quantum algorithms such as quantum optimization and machine…
We study the expressibility and learnability of convex optimization solution functions and their multi-layer architectural extension. The main results are: \emph{(1)} the class of solution functions of linear programming (LP) and quadratic…
Automating algorithm configuration is growing increasingly necessary as algorithms come with more and more tunable parameters. It is common to tune parameters using machine learning, optimizing performance metrics such as runtime and…
We study summation of sequences and integration in the quantum model of computation. We develop quantum algorithms for computing the mean of sequences which satisfy a p-summability condition and for integration of functions from Lebesgue…
Special approximation technique for analysis of different characteristics of states of multipartite infinite-dimensional quantum systems is proposed and applied to study of the relative entropy of entanglement and its regularisation. We…
The main contribution of this dissertation is the introduction of new or improved approximation algorithms and data structures for several similarity search problems. We examine the furthest neighbor query, the annulus query, distance…
We extend molecular bootstrap embedding to make it appropriate for implementation on a quantum computer. This enables solution of the electronic structure problem of a large molecule as an optimization problem for a composite Lagrangian…
We propose a scheme for translating metrological precision bounds into lower bounds on query complexity of quantum search algorithms. Within the scheme the link between quadratic performance enhancement in idealized quantum metrological and…
Random embeddings project high-dimensional spaces to low-dimensional ones; they are careful constructions which allow the approximate preservation of key properties, such as the pair-wise distances between points. Often in the field of…
The question of what can be computed, and how efficiently, are at the core of computer science. Not surprisingly, in distributed systems and networking research, an equally fundamental question is what can be computed in a…
The subspace approximation problem Subspace($k$,$p$) asks for a $k$-dimensional linear subspace that fits a given set of points optimally, where the error for fitting is a generalization of the least squares fit and uses the $\ell_{p}$ norm…
Quantum computers hold promise to improve the efficiency of quantum simulations of materials and to enable the investigation of systems and properties more complex than tractable at present on classical architectures. Here, we discuss…